Integrand size = 13, antiderivative size = 144 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {10 a b^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3} \] Output:
-10*a*b^4*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)-b*sec h(x)^3/(a^2+b^2)/(a+b*sinh(x))+1/3*sech(x)^3*(5*a*b+(a^2-4*b^2)*sinh(x))/( a^2+b^2)^2+1/3*sech(x)*(15*a*b^3+(2*a^4+9*a^2*b^2-8*b^4)*sinh(x))/(a^2+b^2 )^3
Time = 0.32 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.95 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {30 a b^4 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+12 a b^3 \text {sech}(x)-\frac {3 b^5 \cosh (x)}{a+b \sinh (x)}+\left (a^2+b^2\right ) \text {sech}^3(x) \left (2 a b+\left (a^2-b^2\right ) \sinh (x)\right )+\left (2 a^4+9 a^2 b^2-5 b^4\right ) \tanh (x)}{3 \left (a^2+b^2\right )^3} \] Input:
Integrate[Sech[x]^4/(a + b*Sinh[x])^2,x]
Output:
((30*a*b^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 12*a*b^3*Sech[x] - (3*b^5*Cosh[x])/(a + b*Sinh[x]) + (a^2 + b^2)*Sech[x]^3 *(2*a*b + (a^2 - b^2)*Sinh[x]) + (2*a^4 + 9*a^2*b^2 - 5*b^4)*Tanh[x])/(3*( a^2 + b^2)^3)
Time = 0.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3173, 25, 3042, 3345, 25, 3042, 3345, 27, 3042, 3139, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i x)^4 (a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle -\frac {\int -\frac {\text {sech}^4(x) (a-4 b \sinh (x))}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\text {sech}^4(x) (a-4 b \sinh (x))}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {a+4 i b \sin (i x)}{\cos (i x)^4 (a-i b \sin (i x))}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}-\frac {\int -\frac {\text {sech}^2(x) \left (a \left (2 a^2+7 b^2\right )+2 b \left (a^2-4 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\text {sech}^2(x) \left (a \left (2 a^2+7 b^2\right )+2 b \left (a^2-4 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}+\frac {\int \frac {a \left (2 a^2+7 b^2\right )-2 i b \left (a^2-4 b^2\right ) \sin (i x)}{\cos (i x)^2 (a-i b \sin (i x))}dx}{3 \left (a^2+b^2\right )}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}-\frac {\int -\frac {15 a b^4}{a+b \sinh (x)}dx}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {15 a b^4 \int \frac {1}{a+b \sinh (x)}dx}{a^2+b^2}+\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}+\frac {15 a b^4 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}}{3 \left (a^2+b^2\right )}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {30 a b^4 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a^2+b^2}+\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}-\frac {60 a b^4 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\text {sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}(x) \left (\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)+15 a b^3\right )}{a^2+b^2}-\frac {30 a b^4 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}}{3 \left (a^2+b^2\right )}}{a^2+b^2}-\frac {b \text {sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
Input:
Int[Sech[x]^4/(a + b*Sinh[x])^2,x]
Output:
-((b*Sech[x]^3)/((a^2 + b^2)*(a + b*Sinh[x]))) + ((Sech[x]^3*(5*a*b + (a^2 - 4*b^2)*Sinh[x]))/(3*(a^2 + b^2)) + ((-30*a*b^4*ArcTanh[(2*b - 2*a*Tanh[ x/2])/(2*Sqrt[a^2 + b^2])])/(a^2 + b^2)^(3/2) + (Sech[x]*(15*a*b^3 + (2*a^ 4 + 9*a^2*b^2 - 8*b^4)*Sinh[x]))/(a^2 + b^2))/(3*(a^2 + b^2)))/(a^2 + b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Time = 223.30 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.85
| method | result | size |
| default | \(-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a}-b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {5 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 \left (\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-2 a^{3} b -6 a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {2}{3} a^{4}-6 a^{2} b^{2}+\frac {8}{3} b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}-8 a \,b^{3} \tanh \left (\frac {x}{2}\right )^{2}+\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {x}{2}\right )-\frac {2 a^{3} b}{3}-\frac {14 a \,b^{3}}{3}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(266\) |
| risch | \(-\frac {2 \left (-15 a \,b^{4} {\mathrm e}^{7 x}-15 a^{2} b^{3} {\mathrm e}^{6 x}+10 a^{3} b^{2} {\mathrm e}^{5 x}-35 a \,b^{4} {\mathrm e}^{5 x}-10 a^{4} b \,{\mathrm e}^{4 x}-55 a^{2} b^{3} {\mathrm e}^{4 x}+12 a^{5} {\mathrm e}^{3 x}+44 a^{3} b^{2} {\mathrm e}^{3 x}-13 a \,b^{4} {\mathrm e}^{3 x}-4 a^{4} b \,{\mathrm e}^{2 x}-33 a^{2} b^{3} {\mathrm e}^{2 x}+16 b^{5} {\mathrm e}^{2 x}+4 a^{5} {\mathrm e}^{x}+18 a^{3} b^{2} {\mathrm e}^{x}-b^{4} {\mathrm e}^{x} a -2 a^{4} b -9 a^{2} b^{3}+8 b^{5}\right )}{3 \left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}+1\right )^{3} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {5 b^{4} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {5 b^{4} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\) | \(380\) |
Input:
int(sech(x)^4/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-2*b^4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-1/a*b^2*tanh(1/2*x)-b)/(tanh(1/2*x )^2*a-2*b*tanh(1/2*x)-a)-5*a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)- 2*b)/(a^2+b^2)^(1/2)))-2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-a^4-3*a^2*b^2+2* b^4)*tanh(1/2*x)^5+(-2*a^3*b-6*a*b^3)*tanh(1/2*x)^4+(-2/3*a^4-6*a^2*b^2+8/ 3*b^4)*tanh(1/2*x)^3-8*a*b^3*tanh(1/2*x)^2+(-a^4-3*a^2*b^2+2*b^4)*tanh(1/2 *x)-2/3*a^3*b-14/3*a*b^3)/(1+tanh(1/2*x)^2)^3
Leaf count of result is larger than twice the leaf count of optimal. 3044 vs. \(2 (136) = 272\).
Time = 0.14 (sec) , antiderivative size = 3044, normalized size of antiderivative = 21.14 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(sech(x)**4/(a+b*sinh(x))**2,x)
Output:
Integral(sech(x)**4/(a + b*sinh(x))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (136) = 272\).
Time = 0.16 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.40 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {5 \, a b^{4} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, a^{2} b^{3} e^{\left (-6 \, x\right )} - 15 \, a b^{4} e^{\left (-7 \, x\right )} + 2 \, a^{4} b + 9 \, a^{2} b^{3} - 8 \, b^{5} + {\left (4 \, a^{5} + 18 \, a^{3} b^{2} - a b^{4}\right )} e^{\left (-x\right )} + {\left (4 \, a^{4} b + 33 \, a^{2} b^{3} - 16 \, b^{5}\right )} e^{\left (-2 \, x\right )} + {\left (12 \, a^{5} + 44 \, a^{3} b^{2} - 13 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 5 \, {\left (2 \, a^{4} b + 11 \, a^{2} b^{3}\right )} e^{\left (-4 \, x\right )} + 5 \, {\left (2 \, a^{3} b^{2} - 7 \, a b^{4}\right )} e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-5 \, x\right )} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-6 \, x\right )} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-7 \, x\right )} - {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-8 \, x\right )}\right )}} \] Input:
integrate(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
5*a*b^4*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^ 2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2/3*(15*a^2*b ^3*e^(-6*x) - 15*a*b^4*e^(-7*x) + 2*a^4*b + 9*a^2*b^3 - 8*b^5 + (4*a^5 + 1 8*a^3*b^2 - a*b^4)*e^(-x) + (4*a^4*b + 33*a^2*b^3 - 16*b^5)*e^(-2*x) + (12 *a^5 + 44*a^3*b^2 - 13*a*b^4)*e^(-3*x) + 5*(2*a^4*b + 11*a^2*b^3)*e^(-4*x) + 5*(2*a^3*b^2 - 7*a*b^4)*e^(-5*x))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-x) + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*e^(-2*x) + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-3* x) + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-5*x) - 2*(a^6*b + 3*a^4*b ^3 + 3*a^2*b^5 + b^7)*e^(-6*x) + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e ^(-7*x) - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*e^(-8*x))
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (136) = 272\).
Time = 0.14 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.99 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {5 \, a b^{4} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {2 \, {\left (12 \, a b^{3} e^{\left (5 \, x\right )} - 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 8 \, a^{3} b e^{\left (3 \, x\right )} + 32 \, a b^{3} e^{\left (3 \, x\right )} - 6 \, a^{4} e^{\left (2 \, x\right )} - 18 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 12 \, b^{4} e^{\left (2 \, x\right )} + 12 \, a b^{3} e^{x} - 2 \, a^{4} - 9 \, a^{2} b^{2} + 5 \, b^{4}\right )}}{3 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \] Input:
integrate(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
5*a*b^4*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*s qrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2 *(a*b^4*e^x - b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(b*e^(2*x) + 2*a*e ^x - b)) + 2/3*(12*a*b^3*e^(5*x) - 9*a^2*b^2*e^(4*x) + 3*b^4*e^(4*x) + 8*a ^3*b*e^(3*x) + 32*a*b^3*e^(3*x) - 6*a^4*e^(2*x) - 18*a^2*b^2*e^(2*x) + 12* b^4*e^(2*x) + 12*a*b^3*e^x - 2*a^4 - 9*a^2*b^2 + 5*b^4)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(e^(2*x) + 1)^3)
Time = 1.89 (sec) , antiderivative size = 476, normalized size of antiderivative = 3.31 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {8\,\left (a^2-b^2\right )}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {16\,a\,b\,{\mathrm {e}}^x}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,\left (a^6+a^4\,b^2-a^2\,b^4-b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {16\,{\mathrm {e}}^x\,\left (a^5\,b+2\,a^3\,b^3+a\,b^5\right )}{3\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {2\,\left (3\,a^4\,b^2+2\,a^2\,b^4-b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {8\,{\mathrm {e}}^x\,\left (a^3\,b^3+a\,b^5\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (a^2\,b^9+b^{11}\right )}{b^3\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}-\frac {2\,{\mathrm {e}}^x\,\left (a^3\,b^9+a\,b^{11}\right )}{b^4\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {5\,a\,b^4\,\ln \left (-\frac {10\,a\,b^3\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{7/2}}-\frac {10\,a\,b^3\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^3}\right )}{{\left (a^2+b^2\right )}^{7/2}}+\frac {5\,a\,b^4\,\ln \left (\frac {10\,a\,b^3\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{7/2}}-\frac {10\,a\,b^3\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^3}\right )}{{\left (a^2+b^2\right )}^{7/2}} \] Input:
int(1/(cosh(x)^4*(a + b*sinh(x))^2),x)
Output:
((8*(a^2 - b^2))/(3*(a^4 + b^4 + 2*a^2*b^2)) - (16*a*b*exp(x))/(3*(a^4 + b ^4 + 2*a^2*b^2)))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((4*(a^6 - b^ 6 - a^2*b^4 + a^4*b^2))/(a^4 + b^4 + 2*a^2*b^2)^2 - (16*exp(x)*(a*b^5 + a^ 5*b + 2*a^3*b^3))/(3*(a^4 + b^4 + 2*a^2*b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) - ((2*(2*a^2*b^4 - b^6 + 3*a^4*b^2))/(a^4 + b^4 + 2*a^2*b^2)^2 - (8*exp (x)*(a*b^5 + a^3*b^3))/(a^4 + b^4 + 2*a^2*b^2)^2)/(exp(2*x) + 1) - ((2*(b^ 11 + a^2*b^9))/(b^3*(a^2*b + b^3)*(a^2 + b^2)^3) - (2*exp(x)*(a*b^11 + a^3 *b^9))/(b^4*(a^2*b + b^3)*(a^2 + b^2)^3))/(2*a*exp(x) - b + b*exp(2*x)) - (5*a*b^4*log(- (10*a*b^3*(b - a*exp(x)))/(a^2 + b^2)^(7/2) - (10*a*b^3*exp (x))/(a^2 + b^2)^3))/(a^2 + b^2)^(7/2) + (5*a*b^4*log((10*a*b^3*(b - a*exp (x)))/(a^2 + b^2)^(7/2) - (10*a*b^3*exp(x))/(a^2 + b^2)^3))/(a^2 + b^2)^(7 /2)
Time = 0.17 (sec) , antiderivative size = 1067, normalized size of antiderivative = 7.41 \[ \int \frac {\text {sech}^4(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(sech(x)^4/(a+b*sinh(x))^2,x)
Output:
(30*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a* b**5*i + 60*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b **2))*a**2*b**4*i + 60*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sq rt(a**2 + b**2))*a*b**5*i + 180*e**(5*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**4*i + 180*e**(3*x)*sqrt(a**2 + b**2)*ata n((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**4*i - 60*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**5*i + 60*e**x*sqrt(a **2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**4*i - 30*sqrt (a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**5*i - 15*e**(8 *x)*a**2*b**5 - 15*e**(8*x)*b**7 + 30*e**(6*x)*a**4*b**3 - 30*e**(6*x)*b** 7 - 20*e**(5*x)*a**5*b**2 - 40*e**(5*x)*a**3*b**4 - 20*e**(5*x)*a*b**6 + 2 0*e**(4*x)*a**6*b + 130*e**(4*x)*a**4*b**3 + 110*e**(4*x)*a**2*b**5 - 24*e **(3*x)*a**7 - 112*e**(3*x)*a**5*b**2 - 152*e**(3*x)*a**3*b**4 - 64*e**(3* x)*a*b**6 + 8*e**(2*x)*a**6*b + 74*e**(2*x)*a**4*b**3 + 64*e**(2*x)*a**2*b **5 - 2*e**(2*x)*b**7 - 8*e**x*a**7 - 44*e**x*a**5*b**2 - 64*e**x*a**3*b** 4 - 28*e**x*a*b**6 + 4*a**6*b + 22*a**4*b**3 + 17*a**2*b**5 - b**7)/(3*(e* *(8*x)*a**8*b + 4*e**(8*x)*a**6*b**3 + 6*e**(8*x)*a**4*b**5 + 4*e**(8*x)*a **2*b**7 + e**(8*x)*b**9 + 2*e**(7*x)*a**9 + 8*e**(7*x)*a**7*b**2 + 12*e** (7*x)*a**5*b**4 + 8*e**(7*x)*a**3*b**6 + 2*e**(7*x)*a*b**8 + 2*e**(6*x)*a* *8*b + 8*e**(6*x)*a**6*b**3 + 12*e**(6*x)*a**4*b**5 + 8*e**(6*x)*a**2*b...